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%%文档的题目、作者与日期
%\author{王立庆（2021级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
%\title{金融观点下的随机分析基础}
\title{第4章复习题 - 随机分析在金融中的应用}
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{2021 年 9 月 14 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}\itemsep1em

\item  什么是风险资产？解释风险资产的下述模型，
$$X_t = f(t,B_t) = X_0 \exp\left[ \left(c-\frac{1}{2}\sigma^2\right) t +\sigma B_t \right].$$

\vspace{4cm}

\item  什么是相对回报率？什么是波动率？解释下述等式的含义，
$$\frac{X_{t+dt} - X_t}{X_t} = cdt + \sigma dB_t. $$

\vspace{4cm}

\item  什么是无风险收益？解释无风险资产的下述模型，
$$\beta_t =\beta_0e^{rt}. $$

\vspace{4cm}
\newpage

\item  什么是投资组合？什么是交易策略？解释下述表达式的含义，
$$V_t = a_tX_t + b_t\beta_t, \,\,\, t\in [0,T]. $$

\vspace{5cm}

\item  什么是自融资条件？解释自融资条件可以理解为下述等式，
$$dV_t = a_tdX_t + b_t d\beta_t. $$

\vspace{5cm}

\item  什么是欧式看涨期权？什么是敲定价格？解释下述称为未定权益的表达式的含义，
$$(X_T-K)^+ = \max(0, X_T-K) = 
\left\{ \begin{array}{ll} X_T-K, & \text{ if } X_T>K, \\ 0, & \text{ if } X_T\le K. \end{array}\right.$$

\vspace{4cm}
\newpage

\item  Black-Sholes-Merton 是如何定义欧式看涨期权的价格的？

\vspace{9cm}

\item  解释欧式看涨期权的价格的下述模型，
\begin{eqnarray*}
V_t &=& a_tX_t +b_t\beta_t = u(T-t,X_t), \,\,\, t\in [0,T], \\
V_T &=& u(0,X_T)=(X_T-K)^+.
\end{eqnarray*}

\vspace{4cm}
\newpage

\item  解释欧式看涨期权的价格的 Black-Scholes 公式，
$$V_0=u(T,X_0)=X_0\Phi(g(T,X_0)) - Ke^{-rT} \Phi (h(T,X_0)), $$
其中 $g(t,x)$, $h(t,x)$ 和 $\Phi(x)$ 分别定义为
\begin{eqnarray*}
g(t,x) &=& \frac{\ln (x/K) + (r+0.5\sigma^2)t}{\sigma\sqrt{t}}, \\
h(t,x) &=& \frac{\ln (x/K) + (r-0.5\sigma^2)t}{\sigma\sqrt{t}}, \\
\Phi(x) &=& \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{y^2}{2}}dy.
\end{eqnarray*}

\vspace{6cm}

\item  设 $B$ 是标准布朗运动。在什么情况下，下述定义的 $\tilde{B}$ 成为了标准布朗运动？
$$\tilde{B}_t = B_t + qt.$$

\end{enumerate}

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\end{document}

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